A particle is moving along the line y=a with the velocity v = ui, where u is a constant. Express its velocity in plane polar coordinates

To express the velocity in plane polar coordinates, we need to convert the Cartesian coordinates to polar coordinates.

In polar coordinates, a point in the plane is described by its distance r from the origin and the angle θ it makes with the positive x-axis.

The relationship between Cartesian coordinates (x, y) and polar coordinates (r, θ) is given by:

x = r cos(θ) y = r sin(θ)

Given that the particle is moving along the line y = a, we can express this in polar coordinates as:

a = r sin(θ)

The velocity of the particle is given as v = ui. In polar coordinates, the velocity can be decomposed into radial and tangential components. The radial component is the rate of change of the distance from the origin, and the tangential component is the rate of change of the angle.

However, since the particle is moving along a line parallel to the x-axis, its radial velocity is zero and its tangential velocity is u.

Therefore, the velocity of the particle in polar coordinates is v = uθ.